# What Is the correct discounting, risky or riskless?

Suppose I can sell a European put in two ways: 1) in a mark to market collateralized market with collateral rate equal to the riskless rate $r$; 2) in a noncollaterized market where I get the payment for the put up front $0$ and invest in either a risky market account or a risky (defaultable) bond at rate $r+\lambda$.

1) I think the European put price is

$$P_1(t) = \mathbf E\big[e^{-\int_t^T r\,ds}\big(K-S(T)\big)_+\big|\mathcal F_t\big].$$

2) I am of two minds. On the one hand, a riskless instantaneous portfolio could be formed by the put with price $P_2(t)$ and the stock with price $S(t)$, with the portfolio price being $P_2(t)-\frac{\partial P_2(t)}{\partial S}S(t)$, as in the usual Black-Scholes argument. Then the growth rate of this riskless portfolio should be just the riskless rate $r$ and we should just have the same price as in 1). On the other hand, the default probability of the risky bond/money market should impose the credit spread $\lambda$ on the interest rate, so the put price should be $$P_2(t) = \mathbf E\big[e^{-\int_t^T (r+\lambda)\,ds}\big(K-S(T)\big)_+\big|\mathcal F_t\big]$$

Which is the correct discount factor? If neither is true, what is the correct answer?

The discount rate for 2) should be a risky rate $r + \lambda$, although we must talk about how to determine $\lambda$. If you have sold an uncollateralized put option, the put option is an unsecured obligation of yours. Hence it should carry a similar discount rate to other unsecured obligations that you have issued. Thus, $\lambda$ is your credit spread. There is one exception to this: if the buyer of this put option currently owes you money, the premium paid could be considered a reduction of the amount owed, in which case $\lambda$ is HIS credit spread. In no case does $\lambda$ depend on the riskiness of some investment you plan to make with the premium paid. If you are wondering why the Black-Scholes argument breaks down in this case, I believe that it is because the hedged portfolio is risk-free with regard to the stock, but not risk-free with regard to the possibility of default by the issuer of the put option.